Projecting onto intersections of halfspaces and hyperplanes

Hui Ouyang

Published 2020-06-12Version 1

It is well-known that the sequence of iterations of the composition of projections onto affine subspaces converges linearly to the projection onto the intersection of the affine subspaces. Inspired by this, in this work, we systematically study the relations between: projection onto intersection of halfspaces and composition of projections onto halfspaces, projection onto intersection of hyperplane and halfspace and composition of projections onto hyperplane and onto halfspace, and projection onto intersection of hyperplane and halfspace and composition of projections onto halfspace and onto hyperplane. In addition, as by-products, we also provide the Karush-Kuhn-Tucker conditions for characterizing the optimal solution of convex optimization with finitely many equality and inequality constraints in Hilbert spaces and construct a closed formula for the projection onto the intersection of hyperplane and halfspace.